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Yes, your remedy would be against the driver's parents. However, you will probably be at least partially at fault, since it was your daughter who gave them access to the car.
That said, you can't sue them on behalf of your neighbor. Your neighbor is the one with the damaged car, they could sue both parties (you and the driver's parents) or either one of you. In the case that they just want the money from you, you would have to pay it and then try to recoup some of it from the driver.
So yes, you should start by seeing what the driver's parents are willing to contribute. If nothing, you will probably have to sue them. It would be cleaner if the neighbor sued them though.